Looping Conditions for Generalized CollatzSystems

Abstract

We investigate the eventual periodicity of trajectories generated by a family of piecewise-linear maps on the natural numbers that generalize the classical Collatz function. For parameters a, d, m ∈ N, we consider the map

We establish necessary and sufficient conditions on the parameters (a, d, m) for all trajectories to become eventually periodic. In the additive case m = 1, we provide explicit criteria and characterize the structure of eventual cycles when gcd(a, d) = 1. Furthermore, we compute the number of distinct cycles up to orbit equivalence and discuss computational complexity implications.